202 
y\ii. y. s. HOiiGii on tiiii: ai^plication of harmonic 
our knowledge of the tides as they actually exist; but I venture to hope that these 
results, as applied to the oscillations of an ideal ocean, considerably simpler in 
character than the actual ocean, may prove of some interest from the point of view 
of pure hydrodynamical theory. 
In §§ 1—4 I have devoted considerable space to the formation of the dynamical 
equations. The equations obtained agree with those used by Laplace, and 
consequently it may be thought that I have been unnecessarily diffuse over this 
part of tlie subject. My apology is that the questionable, if not erroneous, reasoning 
Avhich has often been assigned for the various approximatioi s introduced seemed to 
me to warrant a very minute examination of the formation of these ecjuations. An 
analytical treatment, such as that I have used, seems to me to be the only safe 
method of procedure to ensure that the approximations do not involve the neglect 
of any terms which may be of equal importance with those retained, many of Avhich 
are extremely small. The method adopted follows liAPLACE in so far as it consists 
of a transformation of the general equations of oscillation of a rotating fluid. I 
trust however that these general equations in the form I have used, which seems 
to be the simplest form to which they can be reduced, may be found less “ repulsive ” 
than those employed by Laplace. 
§ 5 deals with the integration of these ecjuations. The forms of solution discussed 
in the present paper are those Avhich are symmetrical with respect to the axis of 
rotation. The types of oscillation represented by these solutions have been named 
by L aplace, “ Oscillations of the First Species,” but he omitted to discuss them in 
detail on the grounds that the oscillations of such character, which might be 
expected to exist in Nature, would be modified to such an extent by friction that 
they Avould be far better represented by the old “ equilibrium theory,” than by a 
dynamical theory which failed to take due account of the action of friction. The 
tides in question will be of long period, the shortest of the periods being half a lunar 
month in duration, but Professor Darwin Avas, I believe, the first to call attention 
to the fact that this length of period will hardly be sufficiently great to render the 
effects of friction of such paramount importance, and hence he added to the Avork of 
Laplace a discussion of the long-period tides AALen hot subject to frictional 
influences. I have recently attempted to estimate the effects of friction on the tidal 
oscillations of the ocean,and the results at vrhich I have arrhmd fully confirm 
the view of Professor Darwin as to the small influence of friction on the lunar- 
fortnightly tides, and render it highly probable that the effects Avill be almost 
et|ually slight on the solar long-period tides. 
The method of integration I have employed differs from that used by Daraaan, my 
aim having been to express the results by means of series of zonal harmonics instead 
of by the power-series obtained by him. The advantages of this are two-fold; firstly, 
* Ill a paper read before the London Math. Soc., December lOtb, 1896. 
